Approximating the coefficients of the Bessel functions

Abstract: For the type A, BC, and D root systems, we determine equivalent conditions between the coefficients of an exponential holomorphic function and the asymptotic values taken by the Dunkl bilinear form when one of its entries is the function. We establish these conditions over the $|\theta N| \rightarrow\infty$ regime for the type A and D root systems and over the $|\theta_0 N|\rightarrow \infty, \frac{\theta_1}{\theta_0 N}\rightarrow c\in\mathbb{C}$ regime for the type BC root system. We also generalize existing equivalent conditions over the $\theta N \rightarrow c\in\mathbb{C}$ regime for the type A root system and over the $\theta_0 N\rightarrow c_0\in\mathbb{C}, \frac{\theta_1}{\theta_0 N}\rightarrow c_1\in\mathbb{C}$ regime for the type BC root system and prove new equivalent conditions over the $\theta N \rightarrow c\in\mathbb{C}$ regime for the type D root system. Furthermore, we determine the asymptotics of the coefficients of the Bessel functions over the regimes that we have mentioned.